Let me respectfully suggest that the high voltage pulse techique may be a substantial "over-kill" for the spot welding objective.

There are commercial short pulse spot welders.. that use a a cap discharge into a step down transformer... and to make your own is an enjoyable engineering challenge.. but I think there is a much easier and much more accessible apporach to consider.

First.. the amount of energy needed to weld is not large, if it is delivered fairly quickly... few msec.. would be ample.

This suggests a simple power frequency (manual or triac switched) soldering gun, with the following simple modifications.

1. In place of the resistive wire tip used for soldering... you need two movable, conically pointed hefty copper electrodes, between which the pieces to spot welded are clamped. These copper "pole pieces" should be about the same diameter as the main bus bars of the soldering gun and taper to about 1mm diameter, ending with a flat or slightly convex surface.

2. One (or both) of the pole pieces must be movable and able to be clamped against the other pole (with the work between) with a few lbs of force. A simple lever like clamp would work.

Don't have a recent secondary current measurment to quote, but my recollection is the Weller soldering guns deliver at least 150 amps at 60 hz, to the transformer secondary, even at the low heat setting (70W - 100W). So the 140W or the 250W units should be ample to weld almost any wire grids. The use of a series oil filled capacitor in the primary circuit, will boost the current through the transformer, but one needs to be aware that the voltage across the primary will increase.
If you stay below 240VAC, for short bursts, there is no danger whatever, since the magnet wire is typically tested to 1500 v for a few seconds, for UL approved equipment.

3. The last point concerns the controls. I would try a simple push button switch for starters. You only need to get enough heat to melt the wires. The electrode pressure will assure good electrical and thermal contact so the wire will fuse.

There are sophisticated heating cycles that provide annealing currents to give good mechanical properties to the welded joint. This is what the multi-K$$ units provide. One can no doubt approximate these with a little experimentation.

Hope this is helpful.. I think most that communicate here are equal to the task and it is quite safe, since secondary currents while high, are at very low voltages.